If the product of the perpendicular from origin to the pairs of lines x y+x+y+1=0 , x^2-y^2+2 x+1=0 and 2 x^2+3 x y-2 y^2+2 x+1=0 respectively are p₁, p₂ and p₃ , then

Step-by-step solution

Given, x y+x+y+1=0 Comparing above equation with gathered a x^2+b y^2+2 h x y+2 g x+2 f y+c=0 a=0, b=0, h= 1 2 , g= 1 2 , f= 1 2 , c=1 gathered We know that, Product of perpendicular from origin to pair of straight lines is aligned & | c (a-b)^2+4 h^2 | & P₁= | 1 (0-0)^2+4 ( 1 2 )^2 | aligned P₁= 1 1 =1 Now, similarly for x^2-y^2+2 x+1=0 a=1, b=-1, h=0, g=1, f=0, c=1 Then, P₂= | 1 (1+1)^2+4(0)^2 |= 1 (2)^2 = 1 2 and for 2 x^2+3 x y-2 y^2+2 x+1=0 a=1, b=-2, h= 3 2 , g=1, f=0, c=1 Then, n, P₃= | 1 (1+2)^2+4 ( 3 2 )^2